26
G. Altarelli and S. Forte
The second derivatives M 2
ki = (∂ 2 V /∂φ k ∂φ i )(φ i = φ
0
i ) define the squared mass
matrix. Thus the above equation in matrix notation can be written as
M
2 t
A φ
0
= 0 .
(2.52)
In the case of no spontaneous symmetry breaking the ground state is unique, all
symmetry transformations leave it invariant, so that, for all A, t A φ 0 = 0. On the
contrary, if, for some values of A, the vectors (t A φ 0 ) are non-vanishing, i.e. there
is some generator that shifts the ground state into some other state with the same
energy (hence the vacuum is not unique), then each t A φ 0 = 0 is an eigenstate
of the squared mass matrix with zero eigenvalue. Therefore, a massless mode is
associated with each broken generator. The charges of the massless modes (their
quantum numbers in quantum language) differ from those of the vacuum (usually
taken as all zero) by the values of the t A charges: one says that the massless modes
have the same quantum numbers of the broken generators, i.e. those that do not
annihilate the vacuum.
The previous proof of the Goldstone theorem has been given in the classical case.
In the quantum case the classical potential corresponds to tree level approximation
of the quantum potential. Higher order diagrams with loops introduce quantum
corrections. The functional integral formulation of quantum field theory [13, 17]
is the most appropriate framework to define and compute, in a loop expansion,
the quantum potential which specifies, exactly as described above, the vacuum
properties of the quantum theory. If the theory is weakly coupled, e.g. if λ is small,
the tree level expression for the potential is not too far from the truth, and the
classical situation is a good approximation. We shall see that this is the situation
that occurs in the electroweak theory if the Higgs is moderately light (see Chap. 3,
Sect. 3.13.1).
We note that for a quantum system with a finite number of degrees of freedom, for
example one described by the Schrödinger equation, there are no degenerate vacua:
the vacuum is always unique. For example, in the one dimensional Schrödinger
problem with a potential:
V (x) = −μ
2 /2 x
2
+ λ x
4 /4 ,
(2.53)
there are two degenerate minima at x = ±x 0 =
√
(μ 2 /λ) which we denote by
|++ and |−−. But the potential is not diagonal in this basis: the off diagonal matrix
elements:
+|V |−− = =−|V |++ ∼ exp (−khd) = δ
(2.54)
are different from zero due to the non vanishing amplitude for a tunnel effect
between the two vacua, proportional to the exponential of the product of the distance
d between the vacua and the height h of the barrier with k a constant (see Fig. 2.2).
After diagonalization the eigenvectors are (|++ + |−−)/
√
2 and (|++ − |−−)/
√
2,
G. Altarelli and S. Forte
The second derivatives M 2
ki = (∂ 2 V /∂φ k ∂φ i )(φ i = φ
0
i ) define the squared mass
matrix. Thus the above equation in matrix notation can be written as
M
2 t
A φ
0
= 0 .
(2.52)
In the case of no spontaneous symmetry breaking the ground state is unique, all
symmetry transformations leave it invariant, so that, for all A, t A φ 0 = 0. On the
contrary, if, for some values of A, the vectors (t A φ 0 ) are non-vanishing, i.e. there
is some generator that shifts the ground state into some other state with the same
energy (hence the vacuum is not unique), then each t A φ 0 = 0 is an eigenstate
of the squared mass matrix with zero eigenvalue. Therefore, a massless mode is
associated with each broken generator. The charges of the massless modes (their
quantum numbers in quantum language) differ from those of the vacuum (usually
taken as all zero) by the values of the t A charges: one says that the massless modes
have the same quantum numbers of the broken generators, i.e. those that do not
annihilate the vacuum.
The previous proof of the Goldstone theorem has been given in the classical case.
In the quantum case the classical potential corresponds to tree level approximation
of the quantum potential. Higher order diagrams with loops introduce quantum
corrections. The functional integral formulation of quantum field theory [13, 17]
is the most appropriate framework to define and compute, in a loop expansion,
the quantum potential which specifies, exactly as described above, the vacuum
properties of the quantum theory. If the theory is weakly coupled, e.g. if λ is small,
the tree level expression for the potential is not too far from the truth, and the
classical situation is a good approximation. We shall see that this is the situation
that occurs in the electroweak theory if the Higgs is moderately light (see Chap. 3,
Sect. 3.13.1).
We note that for a quantum system with a finite number of degrees of freedom, for
example one described by the Schrödinger equation, there are no degenerate vacua:
the vacuum is always unique. For example, in the one dimensional Schrödinger
problem with a potential:
V (x) = −μ
2 /2 x
2
+ λ x
4 /4 ,
(2.53)
there are two degenerate minima at x = ±x 0 =
√
(μ 2 /λ) which we denote by
|++ and |−−. But the potential is not diagonal in this basis: the off diagonal matrix
elements:
+|V |−− = =−|V |++ ∼ exp (−khd) = δ
(2.54)
are different from zero due to the non vanishing amplitude for a tunnel effect
between the two vacua, proportional to the exponential of the product of the distance
d between the vacua and the height h of the barrier with k a constant (see Fig. 2.2).
After diagonalization the eigenvectors are (|++ + |−−)/
√
2 and (|++ − |−−)/
√
2,